Definition 4.8 (Dominant Distributions)
A distribution P0 dominates another distribution P if
there is a constant cP > 0 such that for all
:

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Definition 4.9 (Universal Distributions)
Let
be a set of probability distributions on
.
A distribution
is universal if for all
:
P0 dominates P.

Theorem 4.2
There is no universal approximable semidistribution.

Proof. The following proof is due to M. Hutter
(personal communications by email following a discussion of
enumerable and approximable universes on 2 August 2000 in
Munich). It is an extension of a modified1 proof [#!LiVitanyi:97!#, p. 249 ff]
that there is no universal recursive semimeasure.

It suffices to focus on .
Identify strings with
integers, and assume P(x) is a universal approximable
semidistribution. We construct an approximable semidistribution
Q(x) that is not dominated by P(x), thus contradicting the
assumption. Let
be a sequence of
recursive functions converging to P(x). We recursively define a
sequence
converging to Q(x). The basic
idea is: each contribution to Q is the sum of n consecutive
P probabilities (n increasing). Define Q0(x):=0;
.
Let n be such that
.
Define jtn (ktn) as the element with
smallest Pt (largest Qt-1) probability in this interval,
i.e.,
.
If
is less than twice and
is more than half of
Qt-1(ktn), set
Qt(x)=Qt-1(x). Otherwise set
for x=jtn and Qt(x)=0 for
.
Qt(x) is
obviously total recursive and non-negative. Since
,
we have

Summing over n we observe that if
Pt is a semidistribution, so is Qt. From some t0 on,
Pt(x) changes by less than a factor of 2 since Pt(x)
converges to
.
Hence Qt(x) remains unchanged for
and converges to
.
But
,
violating our universality assumption
.

Definition 4.10 (Cumulatively Enumerable Distributions - CEDs)
A distribution P on
is a CED
if CP(x) is enumerable for all ,
where